# Consensus Tracking by Iterative Learning Control for Linear Heterogeneous Multiagent Systems Based on Fractional-Power Error Signals

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Digraph

**Assumption**

**1.**

#### 2.2. Critical Definitions

**Definition**

**1.**

**Definition**

**2.**

#### 2.3. Consensus Tracking Problem

**Remark**

**1.**

## 3. Design And Analysis of ILC Consensus Algorithm

Algorithm 1 Iterative learning control (ILC) consensus algorithm based on fractional-power error signals |

Parameters: agent i, iteration index k, ${a}_{ij}$, ${s}_{i}$, ${\mathsf{\Gamma}}_{i}$,${K}_{i}$, $\alpha $ Input: the reference trajectory ${y}_{d}\left(t\right)$ Output: the control input ${u}_{i,k}\left(t\right)$ 1: initialize the iteration index k=1 2: while (${e}_{i,k}\left(t\right)!=0,i\in V,t\in [0,T]$) 3: initialize the initial state ${x}_{i,k}\left(0\right)=0$ 4: calculate the output ${y}_{i,k}\left(t\right)$ by the Equation (2), 5: calculate the tracking errors of each agent ${e}_{i,k}\left(t\right)$ by the Equation (3), 6: calculate the tracking errors with the neighboring agents ${e}_{ij,k}\left(t\right)$ by the Equation (4), 7: calculate ${\xi}_{i,k}\left(t\right)$ by the Equation (5) 8: if k=1 ${u}_{i,k}\left(t\right)=0$ 9: else ${u}_{i,k}\left(t\right)={u}_{i,k-1}\left(t\right)+{\mathsf{\Gamma}}_{i}{\dot{\xi}}_{i,k-1}\left(t\right)+{K}_{i}sign\left({\xi}_{i,k-1}\left(t\right)\right){\left|{\xi}_{i,k-1}\left(t\right)\right|}^{\alpha}$ 10: end if 11: set k=k+1 12: end while 13: print the control input ${u}_{i,k}\left(t\right)$ |

**Remark**

**2.**

**Remark**

**3.**

**Assumption**

**A2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

## 4. Simulation And Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Tracking error of ILC algorithm based on fractional-power error signals with $\alpha =0.7$.

**Figure 7.**Tracking error of ILC algorithm based on fractional-power error signals with $\alpha =0.85$.

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**MDPI and ACS Style**

Luo, Y.-J.; Liu, C.-L.; Liu, G.-Y.
Consensus Tracking by Iterative Learning Control for Linear Heterogeneous Multiagent Systems Based on Fractional-Power Error Signals. *Algorithms* **2019**, *12*, 185.
https://doi.org/10.3390/a12090185

**AMA Style**

Luo Y-J, Liu C-L, Liu G-Y.
Consensus Tracking by Iterative Learning Control for Linear Heterogeneous Multiagent Systems Based on Fractional-Power Error Signals. *Algorithms*. 2019; 12(9):185.
https://doi.org/10.3390/a12090185

**Chicago/Turabian Style**

Luo, Yu-Juan, Cheng-Lin Liu, and Guang-Ye Liu.
2019. "Consensus Tracking by Iterative Learning Control for Linear Heterogeneous Multiagent Systems Based on Fractional-Power Error Signals" *Algorithms* 12, no. 9: 185.
https://doi.org/10.3390/a12090185